 reserve j for set;
 reserve p,r for Real;
 reserve S,T,F for RealNormSpace;
 reserve x0 for Point of S;
 reserve g for PartFunc of S,T;
 reserve c for constant sequence of S;
 reserve R for RestFunc of S,T;
 reserve G for RealNormSpace-Sequence;
 reserve i for Element of dom G;
 reserve f for PartFunc of product G,F;
 reserve x for Element of product G;

theorem Th13:
for G be RealNormSpace-Sequence, p be Point of product G,
    r be Real, r0,p0 be Element of product carr G
  st p=p0
holds r*p = r0
  iff for i be Element of dom G holds r0.i = r*(p0.i)
proof
   let G be RealNormSpace-Sequence, p be Point of product G,
       r be Real, r0,p0 be Element of product carr G;
   assume A1: p=p0;
   hereby assume A2: r*p = r0;
    hereby let i be Element of dom G;
     reconsider i0=i as Element of dom carr G by Lm1;
A3:  (multop G).i0 = the Mult of (G.i0) by PRVECT_2:def 8;
      reconsider rr=r as Element of REAL by XREAL_0:def 1;
     product G = NORMSTR(# product carr G,zeros G,[:addop G:],
       [:multop G:], productnorm G #) by PRVECT_2:6;
     hence r0.i = rr*(p0.i) by A1,A2,A3,PRVECT_2:def 2
           .= r*(p0.i);
    end;
   end;
   assume A4: for i be Element of dom G holds r0.i = r* (p0.i);
   reconsider rp = r*p as Element of product carr G by Th10;
A5:ex g be Function st
      rp = g & dom g = dom carr G
    & for i be object st i in dom carr G holds g.i in (carr G).i
           by CARD_3:def 5;
A6:ex g be Function st
      r0 = g & dom g = dom carr G
    & for i be object st i in dom carr G holds g.i in (carr G).i
           by CARD_3:def 5;
   now let i0 be object;
    assume A7: i0 in dom rp; then
    reconsider i1=i0 as Element of dom G by Lm1,A5;
    reconsider i=i0 as Element of dom carr G by A7,A5;
A8: product G = NORMSTR(# product carr G,zeros G,[:addop G:],
        [:multop G:], productnorm G #) by PRVECT_2:6;
     reconsider r as Element of REAL by XREAL_0:def 1;
    (multop G).i = the Mult of (G.i) by PRVECT_2:def 8; then
    rp.i0 = r*(p0.i1) by A1,A8,PRVECT_2:def 2;
    hence rp.i0 = r0.i0 by A4;
   end;
   hence r*p = r0 by A5,A6,FUNCT_1:2;
end;
